Simulating the non-Hermitian dynamics of financial option pricing with quantum computers

Abstract

The Schrodinger equation describes how quantum states evolve according to the Hamiltonian of the system. For physical systems, we have it that the Hamiltonian must be a Hermitian operator to ensure unitary dynamics. For anti-Hermitian Hamiltonians, the Schrodinger equation instead models the evolution of quantum states in imaginary time. This process of imaginary time evolution has been used successfully to calculate the ground state of a quantum system. Although imaginary time evolution is non-unitary, the normalised dynamics of this evolution can be simulated on a quantum computer using the quantum imaginary time evolution (QITE) algorithm. In this paper, we broaden the scope of QITE by removing its restriction to anti-Hermitian Hamiltonians, which allows us to solve any partial differential equation (PDE) that is equivalent to the Schrodinger equation with an arbitrary, non-Hermitian Hamiltonian. An example of such a PDE is the famous Black-Scholes equation that models the price of financial derivatives. We will demonstrate how our generalised QITE methodology offers a feasible approach for real-world applications by using it to price various European option contracts modelled according to the Black-Scholes equation.

15 pages, 2 figures

References in corpus (9)

• Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution
• Variational ansatz-based quantum simulation of imaginary time evolution
• Pseudo-Supersymmetric Quantum Mechanics and Isospectral Pseudo-Hermi tian Hamiltonians
• Implementation of quantum imaginary-time evolution method on NISQ devices: Nonlocal approximation
• Equivalence of the effective non-hermitian Hamiltonians in the context of open quantum systems and strongly-correlated electron systems
• A variational quantum algorithm for the Feynman-Kac formula
• Efficient Hamiltonian Simulation for Solving Option Price Dynamics
• Efficient quantum imaginary time evolution by drifting real time evolution: an approach with low gate and measurement complexity
• Generalising quantum imaginary time evolution to solve linear partial differential equations